A **4-manifold** is a type of mathematical object studied in the field of topology and differential geometry. In general, an **n-manifold** is a space that locally resembles Euclidean space of dimension \( n \). This means that around every point in a 4-manifold, there exists a neighborhood that is homeomorphic (structurally similar) to an open subset of \( \mathbb{R}^4 \).

In the context of differential geometry and topology, "maps of manifolds" typically refers to smooth or continuous functions that associate points from one manifold to another. Manifolds themselves are mathematical structures that generalize the concept of curves and surfaces to higher dimensions. They can be thought of as "locally Euclidean" spaces, meaning that around any point in a manifold, one can find a neighborhood that looks like Euclidean space.

The term "2-sided" can refer to various concepts depending on the context. Here are a few interpretations: 1. **Physical Objects:** In a physical sense, something that is 2-sided has two distinct sides. This could refer to paper, signs, or any flat object that has a front and a back. 2. **Negotiation:** In the context of negotiation or discussions, a 2-sided approach implies that both parties have the opportunity to express their views, concerns, or proposals.

A 3-manifold is a topological space that locally resembles Euclidean 3-dimensional space. More formally, a space \( M \) is called a 3-manifold if every point in \( M \) has a neighborhood that is homeomorphic (topologically equivalent) to an open subset of \( \mathbb{R}^3 \).

The Loop Theorem, often referred to in the context of topology and knot theory, states that for a given loop (or closed curve) in 3-dimensional space, if the loop does not intersect itself, it can be deformed (or "homotoped") to a simpler form—usually to a point or a standard circle—without leaving the surface it is contained within.

Manifold decomposition is a concept in mathematics and machine learning that involves breaking down complex high-dimensional datasets into simpler, more manageable structures known as manifolds. In this context, a manifold can be understood as a mathematical space that, on a small scale, resembles Euclidean space but may have a more complicated global structure. ### Key Concepts: 1. **Manifolds**: A manifold is a topological space that locally resembles Euclidean space.

The mapping class group is an important concept in the field of algebraic topology, particularly in the study of surfaces and their automorphisms. Specifically, it is the group of isotopy classes of orientation-preserving diffeomorphisms of a surface. Here's a more detailed explanation: 1. **Surface**: A surface is a two-dimensional manifold, which can be either compact (like a sphere, torus, or more complex shapes) or non-compact.

Alexander's trick is a technique used in topology, specifically in the study of continuous functions and compactness. It is primarily associated with the construction of continuous maps and the extension of functions. The trick is named after the mathematician James W. Alexander II and is often employed in scenarios where one needs to extend continuous functions from a subspace to a larger space.

The Alexander horned sphere is a classic example in topology, specifically in the study of knot theory and manifold theory. It is constructed by taking a sphere and creating a complex embedding that demonstrates non-standard behavior in three-dimensional space. The construction of the Alexander horned sphere involves a series of increasingly complicated iterations that result in a space that is homeomorphic to the standard 2-sphere but is not nicely embedded in three-dimensional Euclidean space.

The term "lantern relation" is not widely recognized in most fields, and without additional context, it's challenging to determine its specific meaning. It could refer to a niche concept in a specialized area, or it could be a metaphorical or illustrative term in literature or art.

Geometric topology is a branch of mathematics that focuses on the properties of geometric structures on topological spaces. It combines elements of geometry and topology, investigating spaces that have a geometric structure and understanding how they can be deformed and manipulated. Here is a list of topics that are commonly studied within geometric topology: 1. **Smooth Manifolds**: - Differentiable structures - Tangent bundles - Morse theory 2.

The Blaschke selection theorem is a result in complex analysis and functional analysis concerning the behavior of sequences of Blaschke products, which are a type of analytic function associated with a sequence of points in the unit disk in the complex plane.

The Borromean rings are a set of three interlinked rings that are arranged in such a way that no two rings are directly linked together; instead, all three are interlinked with one another as a complete set. The key property of the Borromean rings is that if any one of the rings is removed, the remaining two rings will be unlinked, meaning they will not be entangled with each other.

The term "boundary parallel" can refer to different concepts depending on the context in which it is used. Generally, it relates to the idea of being aligned or closely associated with the boundaries of a particular system, area, or set of parameters. 1. **In Mathematics and Geometry**: Boundary parallel could describe lines, planes, or surfaces that run parallel to the edges or boundaries of a geometric shape or figure.

Boy's surface is a non-orientable surface that is an example of a mathematical structure in topology. It is a kind of 2-dimensional manifold that cannot be embedded in three-dimensional Euclidean space without self-intersections. Specifically, it can be constructed as a quotient of the 2-dimensional disk, and it can be visualized as a specific kind of "twisted" surface.

Hermann Brunn is not widely recognized in popular culture or historical context, so it's likely that you might be referring to a specific individual or a relatively obscure topic.

The term "crumpled cube" typically refers to a concept in the fields of materials science, mathematics, or physics, commonly associated with the study of shapes and structures. 1. **Materials Science**: In this context, a crumpled cube might study the deformation of materials, particularly how structures like a cube can be manipulated, folded, or crumpled to explore properties such as strength, stability, and energy absorption.

A Dehn twist is a fundamental concept in the field of topology, particularly in the study of surfaces and 3-manifolds. It is a type of homeomorphism that can be used to analyze the properties of surfaces and their mappings.

The Double Suspension Theorem is a concept in algebraic topology, particularly related to the behavior of suspensions in homotopy theory. The theorem provides a relationship between the suspension of a space and the suspension of built spaces from that space.

The Side-Approximation Theorem is a result in non-Euclidean geometry, particularly in the context of hyperbolic geometry. It relates to the conditions under which a triangle can be constructed in hyperbolic space given lengths of the sides.